Neutrinos have been the origin of an impressive number of “surprises”. We know that neutrinos have tiny masses and that oscillations are occurring spontaneously between neutrino species. But additional new discoveries may be ahead of us and they will be discussed in my presentation. Two distinct classes of anomalies have been recently experimentally reported, namely (1) an apparent ≈ 6 % disappearance signal in the antineutrino events detected from near-by nuclear reactors and from the Mega-Curie k-capture calibration sources and (2) observation of an excess signal of unexplained electrons from neutrinos from particle accelerators. Confirming evidence is required. Given that solar neutrino oscillations correspond to ∆m2 ≃ 7×10−5 eV2 and atmospheric neutrino oscillations correspond to ∆m2 ≃ 2.3×10−3 eV2, these anomalies require additional physics. However, the results from the Large Electron Positron collider (LEP) at CERN on the invisible decay width of the Z boson show that there are only three neutrinos with a mass below one half of the mass of the Z boson, which couple to the Z boson and therefore the fourth neutrino, if it indeed exists, cannot couple to the Z boson and hence is a “sterile” neutrino, i.e. a Standard Model gauge singlet. Cosmological data, mainly from observations of the cosmic microwave background and large-scale structure favour the existence of a fourth light degree-of-freedom which could be a “sterile” neutrino.These experiments may all point out to the possible existence of at least one fourth non standard “sterile” neutrino, driving oscillations at small distances, with mass differences of the order of 1 eV2 and relatively large mixing angles. If confirmed, it would imply at least an additional mass-squared difference largely in excess of the Standard Model’s values. Such heavier neutrinos, if discovered, may also contribute to the recent, sofar unexplained cosmological observation of the dark mass excess of non-hadronic origin.Ongoing experimental programmes will be described, which should allow a definitive clarification of the above described “anomalies”.